A case of the Jacobian conjecture
An elementary proof for the following known theorem concerning the Jacobian conjecture is given: Let \(F = (F_ 1, \dots, F_ n) : \mathbb{C}^ n \to \mathbb{C}^ n\) be a polynomial map, \(F_ i = X_ i + H_ i\), \(H_ 1, \dots, H_ n\) homogeneous polynomials in \(\mathbb{C} [X_ 1, \dots, X_ n]\) of degree \(d \geq 2\) and \(J(H)^ 2 = 0\), \(J(H)\) the jacobian matrix of \(H = (H_ 1, \dots, H_ n)\). Then \(F\) has a polynomial \(G\) inverse namely, \(G = (X_ 1 - H_ 1, \dots, X_ n - H_ n)\).
scientific article; zbMATH DE number 681822